A New Arithmetic and an Application to the Computation of Binomial Probability for Very Wide Range of Sample Size

Abstract

To calculate the binomial probability when the sample size is large, many useful approximations without straightforward calculation have been proposed and is often used presently. Faster and more powerful computers are available in the market today that is very cheap. In the present circumstances, question arises that should approximation be used. Among the various kinds of arithmetic which can deal with very large/small number, the most popular one is the multiple-precision arithmetic. In the practical field, for example in order to test hypotheses, we want to get probability quickly without accuracy. In these fields, the multiple-precision arithmetic has demerits such as long calculation time and dependence on the memory size of the personal computer. To avoid these demerits, we propose a new arithmetic on the set of real numbers. Based on the proposed arithmetic, we construct an arithmetic system by C language, which works on the personal computer. This arithmetic system is quite different from the multiple-precision arithmetic. This is applicable to various fields in which numerical calculation is needed. Especially, we try to use this system to calculate the binomial probabilities, because computation of binomial probability by a modern personal computer still includes problems. We propose an algorithm to calculate the binomial probability without approximations and program this algorithm using the proposed arithmetic system. This program enables us to calculate the binomial probability for very large sample size. Numerical experiments show that, from the standpoint of both relative error and calculation time, the proposed algorithm is useful within a very large range of sample size.